Theorem fvreseq 3794

Description: Equality of restricted functions is determined by their values.

Assertion

Ref	Expression
fvreseq	|- (((F Fn A /\ G Fn A) /\ B (_ A) -> ((F |` B) = (G |` B) <-> A.x e. B (F` x) = (G` x)))

Distinct variable groups:   x,B   x,F   x,G

Proof of Theorem fvreseq

Step	Hyp	Ref	Expression
1		fnssres 3596	. . . 4 |- ((F Fn A /\ B (_ A) -> (F |` B) Fn B)
2		fnssres 3596	. . . 4 |- ((G Fn A /\ B (_ A) -> (G |` B) Fn B)
3	1, 2	anim12i 333	. . 3 |- (((F Fn A /\ B (_ A) /\ (G Fn A /\ B (_ A)) -> ((F |` B) Fn B /\ (G |` B) Fn B))
4	3	anandirs 513	. 2 |- (((F Fn A /\ G Fn A) /\ B (_ A) -> ((F |` B) Fn B /\ (G |` B) Fn B))
5		eqfnfv 3792	. . 3 |- (((F |` B) Fn B /\ (G |` B) Fn B) -> ((F |` B) = (G |` B) <-> (B = B /\ A.x e. B ((F |` B)` x) = ((G |` B)` x))))
6		fvres 3729	. . . . . 6 |- (x e. B -> ((F |` B)` x) = (F` x))
7		fvres 3729	. . . . . 6 |- (x e. B -> ((G |` B)` x) = (G` x))
8	6, 7	eqeq12d 1487	. . . . 5 |- (x e. B -> (((F |` B)` x) = ((G |` B)` x) <-> (F` x) = (G` x)))
9	8	ralbiia 1671	. . . 4 |- (A.x e. B ((F |` B)` x) = ((G |` B)` x) <-> A.x e. B (F` x) = (G` x))
10		eqid 1474	. . . . 5 |- B = B
11	10	biantrur 724	. . . 4 |- (A.x e. B ((F |` B)` x) = ((G |` B)` x) <-> (B = B /\ A.x e. B ((F |` B)` x) = ((G |` B)` x)))
12	9, 11	bitr3 175	. . 3 |- (A.x e. B (F` x) = (G` x) <-> (B = B /\ A.x e. B ((F |` B)` x) = ((G |` B)` x)))
13	5, 12	syl6bbr 537	. 2 |- (((F |` B) Fn B /\ (G |` B) Fn B) -> ((F |` B) = (G |` B) <-> A.x e. B (F` x) = (G` x)))
14	4, 13	syl 10	1 |- (((F Fn A /\ G Fn A) /\ B (_ A) -> ((F |` B) = (G |` B) <-> A.x e. B (F` x) = (G` x)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643   (_ wss 2044   |` cres 3168   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  tfrlem1 3906  tfr3 3921  climshft2 7059

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194

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